# Error indicator for the incompressible Darcy flow problems using Enhanced Velocity Mixed Finite Element Method

## TL;DR

This paper develops and compares a posteriori error estimators for the Enhanced Velocity Mixed Finite Element Method in incompressible Darcy flow, emphasizing pressure and velocity error detection to improve mesh adaptivity.

## Contribution

It introduces new residual-based a posteriori error estimators for EVMFEM, including pressure postprocessing, and demonstrates their effectiveness through numerical tests.

## Key findings

- Residual-based estimators outperform explicit residual estimators.
- Pressure postprocessing enhances velocity error detection.
- Numerical results confirm theoretical error bounds.

## Abstract

In the flow and transport numerical simulation, mesh adaptivity strategy is important in reducing the usage of CPU time and memory. The refinement based on the pressure error estimator is commonly-used approach without considering the flux error which plays important role in coupling flow and transport systems. We derive a posteriori error estimators for Enhanced Velocity Mixed Finite Element Method (EVMFEM) in the incompressible Darcy flow. We show numerically difference of the explicit residual based error estimator and implicit error estimators, where Arbogast and Chen post-processing procedure from [1] for pressure was used to improve estimators. A residual-based error estimator provides a better indicator for pressure error. Proposed estimators are good indicators in finding of the large error element. Numerical tests confirm theoretical results. We show the advantage of pressure postprocessing on the detecting of velocity error. To the authors' best knowledge, a posteriori error analysis of EVMFEM has been scarcely investigated from the theoretical and numerical point of view. Reference. 1. Arbogast, T., & Chen, Z. (1995). On the implementation of mixed methods as nonconforming methods for second-order elliptic problems. Mathematics of Computation, 64(211), 943-972.